# Z5 in A5

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z5 and the group is (up to isomorphism) alternating group:A5 (see subgroup structure of alternating group:A5).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Definition

The group is alternating group:A5, which for concreteness, we take as the alternating group on the set .

is the subgroup:

There is a total of five other conjugate subgroups (so six subgroups, including ). The other subgroups are:

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of whole group | 60 | |

order of subgroup | 5 | |

index of a subgroup | 12 | |

size of conjugacy class of subgroup | 6 | |

number of conjugacy classes in automorphism class of subgroup | 1 | |

size of automorphism class of subgroup | 6 |

## GAP implementation

The group-subgroup pair can be constructed as follows:

`G := AlternatingGroup(5); H := Group([(1,2,3,4,5)]);`